A006550
n+8*C(n,2)+30*C(n,3)+62*C(n,4)+75*C(n,5)+30*C(n,6).
Original entry on oeis.org
0, 1, 10, 57, 234, 770, 2136, 5180, 11292, 22599, 42190, 74371, 124950, 201552, 313964, 474510, 698456, 1004445, 1414962, 1956829, 2661730, 3566766, 4715040, 6156272, 7947444, 10153475, 12847926, 16113735, 20043982, 24742684, 30325620
Offset: 1
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 254, gives this as number of ways to color faces of a cube using at most n colors, but the formula is incorrect - see A047780.
- V. Meally, Letter to N. J. A. Sloane, N.D.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).
-
A006550:=(-1-3*z-8*z**2-10*z**3-14*z**4+6*z**5)/(z-1)**7; # conjectured by Simon Plouffe in his 1992 dissertation
-
Table[n+8Binomial[n,2]+30Binomial[n,3]+62Binomial[n,4]+75Binomial[n,5]+ 30Binomial[n,6],{n,0,40}] (* or *) LinearRecurrence[{7,-21,35,-35, 21,-7,1}, {0,1,10,57,234,770,2136},40] (* Harvey P. Dale, Apr 24 2011 *)
A006529
a(n) = (25*n^4-120*n^3+209*n^2-108*n)/6.
Original entry on oeis.org
0, 1, 10, 57, 272, 885, 2226, 4725, 8912, 15417, 24970, 38401, 56640, 80717, 111762, 151005, 199776, 259505, 331722, 418057, 520240, 640101, 779570, 940677, 1125552, 1336425, 1575626, 1845585, 2148832, 2487997, 2865810, 3285101, 3748800, 4259937, 4821642
Offset: 0
- M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246, gives this as the number of ways to color faces of a cube using at most n colors, but the formula is incorrect (it was corrected in the second printing) - see A047780.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
-
A006529:=-z*(1+5*z+17*z**2+77*z**3)/(z-1)**5; [Conjectured by Simon Plouffe in his 1992 dissertation.]
-
Table[(25n^4-120n^3+209n^2-108n)/6,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,1,10,57,272},40] (* Harvey P. Dale, Oct 30 2011 *)
A282816
Number of inequivalent ways to color the faces of a cube using at most n colors so that no two opposite sides have the same color.
Original entry on oeis.org
0, 0, 1, 11, 76, 340, 1135, 3101, 7336, 15576, 30405, 55495, 95876, 158236, 251251, 385945, 576080, 838576, 1193961, 1666851, 2286460, 3087140, 4108951, 5398261, 7008376, 9000200, 11442925, 14414751, 18003636, 22308076, 27437915, 33515185, 40674976, 49066336
Offset: 0
For n = 2 we get a(2) = 1 way to color the faces of a cube with two colors so that no two opposite sides have the same color.
-
Table[(8n(n-1) + n^3(n-1)^3) /24, {n, 0, 35}]
-
a(n) = n*(n-1)*(n^4-2*n^3+n^2+8)/24 \\ Charles R Greathouse IV, Feb 22 2017
A282817
Number of inequivalent ways to color the faces of a cube using at most n colors so that no color appears more than twice.
Original entry on oeis.org
0, 0, 0, 6, 72, 375, 1320, 3675, 8736, 18522, 36000, 65340, 112200, 184041, 290472, 443625, 658560, 953700, 1351296, 1877922, 2565000, 3449355, 4573800, 5987751, 7747872, 9918750, 12573600, 15795000, 19675656, 24319197, 29841000, 36369045, 44044800, 53024136
Offset: 0
For n=3 we get a(3)=6 ways to color the faces of a cube with three colors so that no color appears more than twice.
-
Table[(3 n (n - 1) (n - 2)^2 + 6 n (n - 1) (n - 2) + n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) + 15 n (n - 1) (n - 2) (n - 3) (n - 4) + 45 n (n - 1) (n - 2) (n - 3) + 15 n (n - 1) (n - 2))/24, {n, 0, 16}]
A316093
Non-isomorphic colorings of the cube under rotations, using at most N colors on the faces and M colors on the vertices. Square array H(N,M) with N,M > 0 read by antidiagonals.
Original entry on oeis.org
1, 10, 23, 57, 776, 333, 240, 8121, 17946, 2916, 800, 44608, 200961, 176160, 16725, 2226, 168675, 1124208, 1995852, 1045050, 70911, 5390, 501528, 4281300, 11198720, 11877825, 4485960, 241913, 11712, 1261701, 12773538, 42697300, 66700400, 51044337, 15385706, 701968, 23355, 2807296, 32195646, 127461216, 254387500, 286724160, 175153881, 44761216, 1798281, 43450, 5685903, 71718336, 321364540, 759518850, 1093653675, 983988208, 509689776, 114826410, 4173775
Offset: 1
Square array begins:
1, 10, 57, 240, 800, ...
23, 776, 8121, 44608, 168675, ...
333, 17946, 200961, 1124208, 4281300, ...
2916, 176160, 1995852, 11198720, 42697300, ...
Comments